Crystal structure

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter; the smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure. The unit cell reflects the symmetry and structure of the entire crystal, built up by repetitive translation of the unit cell along its principal axes; the translation vectors define the nodes of the Bravais lattice. The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants called lattice parameters or cell parameters; the symmetry properties of the crystal are described by the concept of space groups. All possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups.

The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage, electronic band structure, optical transparency. Crystal structure is described in terms of the geometry of arrangement of particles in the unit cell; the unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a parallelepiped, providing six lattice parameters taken as the lengths of the cell edges and the angles between them; the positions of particles inside the unit cell are described by the fractional coordinates along the cell edges, measured from a reference point. It is only necessary to report the coordinates of a smallest asymmetric subset of particles; this group of particles may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell.

The collection of symmetry operations of the unit cell is expressed formally as the space group of the crystal structure. Vectors and planes in a crystal lattice are described by the three-value Miller index notation; this syntax uses the indices ℓ, m, n as directional orthogonal parameters, which are separated by 90°. By definition, the syntax denotes a plane that intercepts the three points a1/ℓ, a2/m, a3/n, or some multiple thereof; that is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell. If one or more of the indices is zero, it means. A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined; the Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in. In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Considering only planes intersecting one or more lattice points, the distance d between adjacent lattice planes is related to the reciprocal lattice vector orthogonal to the planes by the formula d = 2 π | g ℓ m n | The crystallographic directions are geometric lines linking nodes of a crystal.

The crystallographic planes are geometric planes linking nodes. Some directions and planes have a higher density of nodes; these high density planes have an influence on the behavior of the crystal as follows: Optical properties: Refractive index is directly related to density. Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules; these phenomena are thus sensitive to the density of nodes. Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species; the surface tension of an interface thus varies according to the density on the surface. Microstructural defects: Pores and crystallites tend to have straight grain boundaries following higher density planes. Cleavage: This occurs preferentially parallel to higher density planes. Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes; the perturbation carried by the dislocation is along a dense direction.

The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice. Some directions and planes are defined by symmetry of the crystal system. In monoclinic, rhombohedral and trigonal/hexagonal systems there is one unique axis which has higher rotational symmetry than the other two axes; the basal plane is the plane perpendicular to the principal axis in these crystal systems. For triclinic and cubic crystal systems the axis designation is arbitrary and there is no principal axis. For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length. So, in this common case, the Miller indices and both denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent l

Basis set (chemistry)

A basis set in theoretical and computational chemistry is a set of functions, used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer. The use of basis sets is equivalent to the use of an approximate resolution of the identity; the single-particle states are expressed as linear combinations of the basis functions. The basis set can either be composed of atomic orbitals, the usual choice within the quantum chemistry community, or plane waves which are used within the solid state community. Several types of atomic orbitals can be used: Gaussian-type orbitals, Slater-type orbitals, or numerical atomic orbitals. Out of the three, Gaussian-type orbitals are by far the most used, as they allow efficient implementations of Post-Hartree–Fock methods. In modern computational chemistry, quantum chemical calculations are performed using a finite set of basis functions.

When the finite basis is expanded towards an complete set of functions, calculations using such a basis set are said to approach the complete basis set limit. In this article, basis function and atomic orbital are sometimes used interchangeably, although it should be noted that the basis functions are not true atomic orbitals, because many basis functions are used to describe polarization effects in molecules. Within the basis set, the wavefunction is represented as a vector, the components of which correspond to coefficients of the basis functions in the linear expansion. One-electron operators correspond to matrices, in this basis, whereas two-electron operators are rank four tensors; when molecular calculations are performed, it is common to use a basis composed of atomic orbitals, centered at each nucleus within the molecule. The physically best motivated basis set are Slater-type orbitals, which are solutions to the Schrödinger equation of hydrogen-like atoms, decay exponentially far away from the nucleus.

While hydrogen-like atoms lack many-electron interactions, it can be shown that the molecular orbitals of Hartree-Fock and density-functional theory exhibit exponential decay. Furthermore, S-type STOs satisfy Kato's cusp condition at the nucleus, meaning that they are able to describe electron density near the nucleus. However, calculating integrals with STOs is computationally difficult and it was realized by Frank Boys that STOs could be approximated as linear combinations of Gaussian-type orbitals instead; because the product of two GTOs can be written as a linear combination of GTOs, integrals with Gaussian basis functions can be written in closed form, which leads to huge computational savings. Dozens of Gaussian-type orbital basis sets have been published in the literature. Basis sets come in hierarchies of increasing size, giving a controlled way to obtain more accurate solutions, however at a higher cost; the smallest basis sets are called minimal basis sets. A minimal basis set is one in which, on each atom in the molecule, a single basis function is used for each orbital in a Hartree–Fock calculation on the free atom.

For atoms such as lithium, basis functions of p type are added to the basis functions that correspond to the 1s and 2s orbitals of the free atom, because lithium has a 1s2p bound state. For example, each atom in the second period of the periodic system would have a basis set of five functions; the minimal basis set is close to exact for the gas-phase atom. In the next level, additional functions are added to describe polarization of the electron density of the atom in molecules; these are called polarization functions. For example, while the minimal basis set for hydrogen is one function approximating the 1s atomic orbital, a simple polarized basis set has two s- and one p-function; this adds flexibility to the basis set allowing molecular orbitals involving the hydrogen atom to be more asymmetric about the hydrogen nucleus. This is important for modeling chemical bonding, because the bonds are polarized. D-type functions can be added to a basis set with valence p orbitals, f-functions to a basis set with d-type orbitals, so on.

Another common addition to basis sets is the addition of diffuse functions. These are extended Gaussian basis functions with a small exponent, which give flexibility to the "tail" portion of the atomic orbitals, far away from the nucleus. Diffuse basis functions are important for describing anions or dipole moments, but they can be important for accurate modeling of intra- and intermolecular bonding; the most common minimal basis set is STO-nG. This n value represents the number of Gaussian primitive functions comprising a single basis function. In these basis sets, the same number of Gaussian primitives comprise valence orbitals. Minimal basis sets give rough results that are insufficient for research-quality publication, but are much cheaper than their larger counterparts. Used minimal basis sets of this type are: STO-3G STO-4G STO-6G STO-3G* - Polarized version of STO-3GThere are several other minimum basis sets that have been used such as the MidiX basis sets. During most molecular bonding, it is the valence electrons which principally take part in the bonding.

In recognition of this fact, it is common to represent valence orbitals by more than one basis fun